Senate Majority Leader Mitch McConnell said Tuesday that the Senate will only be debating immigration this week, and said if Democrats were serious about trying to find a solution for illegal immigrant “Dreamers” they better begin to offer plans.

“We’ll need to wrap this up this week,” Mr. McConnell said. “If there are 60 votes for any of these proposals here in the Senate, we should be able to discover that.”

Senators are slated to be on a Washington’s birthday vacation next week, helping enforce Mr. McConnell’s firm deadline.

Democrats have yet to offer any firm legislative language to deal with Dreamers, though Minority Leader Charles E. Schumer said earlier in the day that his party will back a proposal to grant a generous amnesty for as many as 3 million illegal immigrants in exchange for a border security study and promises of future changes.

Mr. Schumer wants the legalization debate to encompass as many illegal immigrants as possible, but said he doesn’t want to go beyond “border security” on the enforcement side, which is why he objected to votes on a sanctuary city crackdown earlier Tuesday.